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Prove the following statement. "The bise...

Prove the following statement. "The bisector of an angle of a triangle divides the sides opposite to the angle in the ratio of the remaining sides"

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Statement : The bisector of an angle of a triangle divides the side opposite to the angle in the ratio of the remaining sides.
Given In `Delta PQR`, bisector of `/_ PQR` intersects side PR in point S such that P-S-R
To prove : `(PQ)/(QR) = (PS)/(SR)`
(`(1)/(2)` mark of figure `(1)/(2)` mark for given, to prove and construction)
Construction : Draw a line parallel to ray QS through point R. Extend seg PQ to intersect the line at point M such that P-Q-M.
Proof : In `Delta PMR`
seg, QS||side MR .....[Construction]
`:.` by basic proportionality theorem,
`(PQ)/(QM = (PS)/(SR)`....(1)
Ray QS||side MR and line PM is the transversal
`:. /_ PQS ~= /_ QMR`...... [Corresponding angles]
Ray QS||side MR and line QR is the transversal.
`:. /_ SQR ~= /_ QRM` ....[Alternate angles]
`/_ PQS ~= /_SQR`
...[Ray QS is the bisector of `/_ PQR` ...(4)
In `Delta QRM`
`/_ QMR ~= /_ QRM`..... [From (2),(3) and (4)]
`:.` seg QR `~=` seg QM ,
...[Converse of isosceles triangle theorem]
`:. (PQ)/(QR) = (PS)/(SR)` .....[Form (1) and (5)]
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